Workshop on Extremal Graph Theory

The Yandex workshop on extremal graph theory is part of the 9th International Computer Science Symposium and aims at brining together specialists who work in extremal combinatorics, probabilistic methods in discrete mathematics, random graphs, hypergraphs and related fields.

Speakers in this workshop, which is organised by Andrey Raigorodsky, include David Gamarnik (MIT), Joel Spencer (Courant Institute, New York University), Endre Szemerédi (Rutgers University) and others.

This workshop will take place in Yandex’s Moscow office on June 6, 2014.

The talks will start at 10:00 am with on-site registration open from 9:30 am.

To be eligible for on-site registration, participants fist need to register online.

Workshop on Extremal Graph Theory

Finding Needles in Exponential Haystacks

We discuss two recent methods in which an object with a certain property
is sought. In both, using of a straightforward random object would succeed
with only exponentially small probability. The new randomized algorithms
run efficiently and also give new proofs of the existence of the desired object.
In both cases there is a potentially broad use of the methodology.

(i) Consider an instance of k-SAT in which each clause overlaps (has
a variable in common, regardless of the negation symbol) with at most d
others. Lovasz showed that when ed < 2k (regardless of the number of
variables) the conjunction of the clauses was satisfiable. The new approach
due to Moser is to start with a random true-false assignment. In a WHILE
loop, if any clause is not satisfied we ”fix it” by a random reassignment. The
analysis of the algorithm is unusual, connecting the running of the algorithm
with certain Tetris patterns, and leading to some algebraic combinatorics.
[These results apply in a quite general setting with underlying independent
”coin flips” and bad events (the clause not being satisfied) that depend on
only a few of the coin flips.]

(ii) No Outliers. Given n vectors rj in n-space with all coefficients in
[−1,+1] one wants a vector x = (x1, ..., xn) with all xi = +1 or −1 so that all dot products x · rj are at most K
√n in absolute value, K an absolute constant. A random x would make x · rj Gaussian but there would be outliers. The existence of such an x was first shown by the speaker. The first algorithm was found by Nikhil Bansal. The approach here, due to Lovett and Meka, is to begin with x = (0, ..., 0) and let it float in a kind of restricted Brownian Motion until all the coordinates hit the boundary.

Workshop on Extremal Graph Theory

Zero-one k-laws for G(n,n−α)

We study asymptotical behavior of the probabilities of first-order
properties for Erdős-Rényi random graphs G(n,p(n)) with
p(n)=n-α, α ∈ (0,1). The following zero-one law
was proved in 1988 by S. Shelah and J.H. Spencer [1]: if
α is irrational then for any first-order property L
either the random graph satisfies the property L asymptotically
almost surely or it doesn't satisfy (in such cases the random
graph is said to obey zero-one law. When α ∈ (0,1)
is rational the zero-one law for these graphs doesn't hold.

Let k be a positive integer. Denote by Lk the class
of the first-order properties of graphs defined by formulae with
quantifier depth bounded by the number k (the sentences are of a
finite length). Let us say that the random graph obeys
zero-one k-law, if for any first-order property
L ∈ Lk either the random graph satisfies the property
L almost surely or it doesn't satisfy. Since 2010 we prove
several zero-one $k$-laws for rational α from
Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some
points from Ik we disprove the law. In particular, for
α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one
k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)},
then zero-one law does not hold (in such cases we call the number
αk-critical).

We also disprove the law for some
α ∈ [2/(k-1), k/(k+1)]. From our results it
follows that zero-one 3-law holds for any α ∈ (0,1).
Therefore, there are no 3-critical points in (0,1). Zero-one
4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2
and 13/14 are 4-critical. Moreover, we know some rational
4-critical and not 4-critical numbers in [7/8,13/14). The
number 2/3 is 4-critical. Recently we obtain new results
concerning zero-one 4-laws for the neighborhood of the number
2/3.

Workshop on Extremal Graph Theory

Improved algorithms for colorings of simple hypergraphs and applications

The famous Lovász Local Lemma was derived in the paper of P. Erdős and Lovász to prove that any n-uniform non-r-colorable hypergraph H has maximum edge degree at least

Δ(H) ≥ ¼ rn−1.

A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs.

In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple n-uniform non-r-colorable hypergraph H has maximum edge degree at least

Δ(H) ≥ с · nrn−1

where c > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the b-simple case.

The work of the second author was supported by Russian Foundation of Fundamental Research (grant № 12-01-00683-a), by the program “Leading Scientific Schools” (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1

Workshop on Extremal Graph Theory

Proof of the Pósa−Seymour Conjecture

In 1974 Paul Seymour conjectured that any graph G of order n and minimum degree at least (k−1)/k · n contains the (k − 1)th power of a Hamiltonian cycle. This conjecture was proved with the help of the Regularity Lemma – Blow-up Lemma method for n ≥ n0 where n0 is very large. Here we present another proof that avoids the use of the Regularity Lemma and thus the resulting n0 is much smaller. The main ingredient is a new kind of connecting lemma.

Workshop on Extremal Graph Theory

Studying zero-one laws for random graphs was initiated by Glebskii Y. et al. in [1]. In this work the authors proved the zero-one law for Erdős-Rényi random graph G(n,p). Later S. Shelah and J. Spencer expanded the class of functions p(n), for which G(n,p) follows the zero-one law (see [2]). Zero-one laws for random distance graphs have been considered for the rst time by M. Zhukovskii (see [3]). In [4] we studied the zero-one law for a more general model of random distance graphs.

Let {Gn = (Vn, En)}∞n=1 be a a sequence of distance graphs and p = p(n) be
a function of n. The random distance graph G(Gn,p) is the probabilistic space (ΩGn,FGn,ΡGnρ), where

ΩGn = {G = (V,E) : V = Vn, E ⊆ En},

FGn = 2 ΩGn, FGn,p(G) = p|E|(1-p)|En|-|E|.

We say sequence G(Gn, p) follows zero-one k-law if for any first-order property L with quantier depth at most k the probability PGn ,p(L) of the event "G(Gn,p)possesses property L" tends either to 0 or to 1 as n → ∞. We say sequence G(Gn, p) follows extended zero-one k-law if for any first-order property L with quantier depth at most k any partial limit of the sequence {PGn,p(L)}∞n=1 equals either 0 or 1.

We obtain conditions on the sequence {Gn}∞n=1 under which one of the following three mutually exclusive cases occurs:

Workshop on Extremal Graph Theory

Subsets of Z/pZ with small Wiener norm and arithmetic progressions

It is proved that any subset of Z/pZ, p is a prime number, having
small Wiener norm (l_1-norm of its Fourier transform) contains a
subset which is close to be an arithmetic progression. We apply the
obtained results to get some progress in so-called Littlewood
conjecture in Z/pZ as well as in a quantitative version of
Beurling-Helson theorem.

Workshop on Extremal Graph Theory

Limits of Local Algorithms for Randomly Generated Constraint Satisfaction Problems

A major challenge in the field of random graphs is constructing fast algorithms for solving a variety of combinatorial optimization problems, such
as finding largest independent set of a graph or finding a satisfying assignment in random instances of K-SAT problem. Most of the algorithms
that have been successfully analyzed in the past are so-called local algorithms which rely on making decisions based on local information.

In this talk we will discuss fundamental barrier on the power of local algorithms to solve such problems, despite the conjectures put forward in the past. In particular, we refute a conjecture regarding the power of local algorithms to find nearly largest independent sets in random regular graphs. Similarly, we show that a broad class of local algorithms, including the so-called Belief Propagation and Survey Propagation algorithms, cannot find satisfying assignments in random NAE-K-SAT problem above a certain asymptotic threshold, below which even simple algorithms succeed with high probability. Our negative results exploit fascinating geometry of feasible solutions of random constraint satisfaction problems, which was first predicted by physicists heuristically and now confirmed by rigorous methods. According to this picture, the solution space exhibits a clustering property whereby the feasible solutions tend to cluster according to the underlying Hamming distance. We show that success of local algorithms would imply violation of such a clustering property thus leading to a contradiction.

Local clustering coefficient in preferential attachment graphs

In this talk we discuss some properties of generalized preferential attachment models. A general approach to preferential attachment was introduced in [1], where a wide class of models (PA-class) was defined in terms of constraints that are sufficient for the study of the degree distribution and the clustering coefficient.

It was shown in [1] that the degree distribution in all models of the PA-class follows the power law. Also, the global clustering coefficient was analyzed and a lower bound for the average local clustering coefficient was obtained. It was also shown that in preferential attachment models global and average local clustering coefficients behave differently.

In our study we expand the results of [1] by analyzing the local clustering coefficient for the PA-class of models. We analyze the behavior of C(d) which is the average local clustering for vertices of degree d. The value C(d) is defined in the following way. First, the local clustering of a given vertex is defined as the ratio of the number of edges between the neighbors of this vertex to the number of pairs of such neighbors. Then the obtained values are averaged over all vertices of degree d.

Workshop on Extremal Graph Theory

There is a vast empirical research on the behaviour of ranking algorithms, e.g. Google PageRank, in scale-free networks. In this talk, we address this problem by analytical probabilistic methods. In particular, it is well-known that the PageRank in scale-free networks follows a power law with the same exponent as in-degree. Recent probabilistic analysis has provided an explanation for this phenomenon by obtaining a natural approximation for PageRank based on stochastic fixed-point equations. For these equations, explicit solutions can be constructed on weighted branching trees, and their tail behavior can be described in great detail.

In this talk we present a model for generating directed random graphs with prescribed degree distributions where we can prove that the PageRank of a randomly chosen node does indeed converge to the solution of the corresponding fixed-point equation as the number of nodes in the graph grows to infinity. The proof of this result is based on classical random graph coupling techniques combined with the now extensive literature on the behavior of branching recursions on trees.

Workshop on Extremal Graph Theory

Boundary properties of factorial classes of graphs

For a class of graphs X, let X_n be the number of graphs with vertex set {1,...,n} in the class X, also known as the speed of X. It is known that in the family of hereditary classes (i.e. those that are closed under taking induced subgraphs) the speeds constitute discrete layers and the first four lower layers are constant, polynomial, exponential, and factorial. For each of these four layers a complete list of minimal classes is available, and this information allows to provide a global structural characterization for the first three of them. The minimal layer for which no such characterization is known is the factorial one. A possible approach to obtaining such a characterization could be through identifying all minimal superfactorial classes. However, no such class is known and possibly no such class exists. To overcome this difficulty, we employ the notion of boundary classes that has been recently introduced to study algorithmic graph problems and reveal the first few boundary classes for the factorial layer.